Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 1 Serway/Jewett, Physics for Scientists and Engineers, 8/e PSE 8e - Chapter 15 Oscillatory Motion Questions and Problems Objective Questions denotes answer available in Student Solutions Manual / Study Guide 1. The position of an object moving with simple harmonic motion is given by x 4cos 6 t , where x is in meters and t is in seconds. What is the period of the oscillating system? (a) 4 s (b) 1 6 s (c) 1 3 s (d) 6 s (e) impossible to determine from the information given 2. Which of the following statements is not true regarding a mass–spring system that moves with simple harmonic motion in the absence of friction? (a) The total energy of the system remains constant. (b) The energy of the system is continually transformed between kinetic and potential energy. (c) The total energy of the system is proportional to the square of the amplitude. (d) The potential energy stored in the system is greatest when the mass passes through the equilibrium position. (e) The velocity of the oscillating mass has its maximum value when the mass passes through the equilibrium Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 2 position. 3. A block–spring system vibrating on a frictionless, horizontal surface with an amplitude of 6.0 cm has an energy of 12 J. If the block is replaced by one whose mass is twice the mass of the original block and the amplitude of the motion is again 6.0 cm, what is the energy of the system? (a) 12 J (b) 24 J (c) 6 J (d) 48 J (e) none of those answers 4. If an object of mass m attached to a light spring is replaced by one of mass 9m, the frequency of the vibrating system changes by what factor? (a) 1 9 (b) 1 3 (c) 3.0 (d) 9.0 (e) 6.0 5. An object of mass 0.40 kg, hanging from a spring with a spring constant of 8.0 N/m, is set into an up-and-down simple harmonic motion. What is the magnitude of the acceleration of the object when it is at its maximum displacement of 0.10 m? (a) zero (b) 0.45 m/s2 (c) 1.0 m/s2 (d) 2.0 m/s2 (e) 2.4 m/s2 6. A runaway railroad car, with mass 3.0 105 kg, coasts across a level track at 2.0 m/s when it collides elastically with a spring-loaded bumper at the end of the track. If the spring constant of the bumper is 2.0 106 N/m, what is the maximum compression of the spring during the collision? (a) 0.77 m (b) 0.58 m (c) 0.34 m (d) 1.07 m (e) 1.24 m Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 3 7. If a simple pendulum oscillates with small amplitude and its length is doubled, what happens to the frequency of its motion? (a) It doubles. (b) It becomes 2 times as large. (c) It becomes half as large. (d) It becomes 1/ 2 times as large. (e) It remains the same. 8. An object–spring system moving with simple harmonic motion has an amplitude A. When the kinetic energy of the object equals twice the potential energy stored in the spring, what is the position x of the object? (a) A (b) 1 3 A (c) A 3 (d) 0 (e) none of those answers 9. A particle on a spring moves in simple harmonic motion along the x axis between turning points at x1 = 100 cm and x2 = 140 cm. (i) At which of the following positions does the particle have maximum speed? (a) 100 cm (b) 110 cm (c) 120 cm (d) at none of those positions (ii) At which position does it have maximum acceleration? Choose from the same possibilities as in part (i). (iii) At which position is the greatest net force exerted on the particle? Choose from the same possibilities as in part (i). 10. A mass–spring system moves with simple harmonic motion along the x axis between turning points at x1 = 20 cm and x2 = Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 4 60 cm. For parts (i) through (iii), choose from the same five possibilities. (i) At which position does the particle have the greatest magnitude of momentum? (a) 20 cm (b) 30 cm (c) 40 cm (d) some other position (e) The greatest value occurs at multiple points. (ii) At which position does the particle have greatest kinetic energy? (iii) At which position does the particle-spring system have the greatest total energy? 11. A block with mass m = 0.1 kg oscillates with amplitude A = 0.1 m at the end of a spring with force constant k = 10 N/m on a frictionless, horizontal surface. Rank the periods of the following situations from greatest to smallest. If any periods are equal, show their equality in your ranking. (a) The system is as described above. (b) The system is as described in situation (a) except the amplitude is 0.2 m. (c) The situation is as described in situation (a) except the mass is 0.2 kg. (d) The situation is as described in situation (a) except the spring has force constant 20 N/m. (e) A small resistive force makes the motion underdamped. 12. For a simple harmonic oscillator, answer yes or no to the following questions. (a) Can the quantities position and velocity have the same sign? (b) Can velocity and acceleration have the same sign? (c) Can position and acceleration have the same sign? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 5 13. The top end of a spring is held fixed. A block is hung on the bottom end as in Figure OQ15.13a, and the frequency f of the oscillation of the system is measured. The block, a second identical block, and the spring are carried up in a space shuttle to Earth orbit. The two blocks are attached to the ends of the spring. The spring is compressed without making adjacent coils touch (Fig. OQ15.13b), and the system is released to oscillate while floating within the shuttle cabin (Fig. OQ15.13c). What is the frequency of oscillation for this system in terms of f? (a) f/2 (b) f / 2 (c) f (d) 2 f (e) 2f 14. You attach a block to the bottom end of a spring hanging vertically. You slowly let the block move down and find that it hangs at rest with the spring stretched by 15.0 cm. Next, you lift the block back up to the initial position and release it from rest with the spring unstretched. What maximum distance does it move down? (a) 7.5 cm (b) 15.0 cm (c) 30.0 cm (d) 60.0 cm (e) The distance cannot be determined without knowing the mass Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 6 and spring constant. 15. A simple pendulum has a period of 2.5 s. (i) What is its period if its length is made four times larger? (a) 1.25 s (b) 1.77 s (c) 2.5 s (d) 3.54 s (e) 5 s (ii) What is its period if the length is held constant at its initial value and the mass of the suspended bob is made four times larger? Choose from the same possibilities. 16. A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is determined. (i) When the elevator accelerates upward, is the period (a) greater, (b) smaller, or (c) unchanged? (ii) When the elevator has a downward acceleration, is the period (a) greater, (b) smaller, or (c) unchanged? (iii) When the elevator moves with constant upward velocity, is the period of the pendulum (a) greater, (b) smaller, or (c) unchanged? 17. You stand on the end of a diving board and bounce to set it into oscillation. You find a maximum response in terms of the amplitude of oscillation of the end of the board when you bounce at frequency f. You now move to the middle of the board and repeat the experiment. Is the resonance frequency for forced oscillations at this point (a) higher, (b) lower, or (c) the same as f? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 7 Conceptual Questions denotes answer available in Student Solutions Manual / Study Guide 1. Is a bouncing ball an example of simple harmonic motion? Is the daily movement of a student from home to school and back simple harmonic motion? Why or why not? 2. The equations listed in Table 2.2 give position as a function of time, velocity as a function of time, and velocity as a function of position for an object moving in a straight line with constant acceleration. The quantity vxi appears in every equation. (a) Do any of these equations apply to an object moving in a straight line with simple harmonic motion? (b) Using a similar format, make a table of equations describing simple harmonic motion. Include equations giving acceleration as a function of time and acceleration as a function of position. State the equations in such a form that they apply equally to a block–spring system, to a pendulum, and to other vibrating systems. (c) What quantity appears in every equation? 3. (a) If the coordinate of a particle varies as x = –A cos t, what is the phase constant in Equation 15.6? (b) At what position is the particle at t = 0? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 8 4. A simple pendulum can be modeled as exhibiting simple harmonic motion when is small. Is the motion periodic when is large? 5. Figure CQ15.5 shows graphs of the potential energy of four different systems versus the position of a particle in each system. Each particle is set into motion with a push at an arbitrarily chosen location. Describe its subsequent motion in each case (a), (b), (c), and (d). 6. A student thinks that any real vibration must be damped. Is the student correct? If so, give convincing reasoning. If not, give an example of a real vibration that keeps constant amplitude forever if the system is isolated. 7. The mechanical energy of an undamped block–spring system is constant as kinetic energy transforms to elastic potential energy and vice versa. For comparison, explain what happens Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 9 to the energy of a damped oscillator in terms of the mechanical, potential, and kinetic energies. 8. Is it possible to have damped oscillations when a system is at resonance? Explain. 9. Will damped oscillations occur for any values of b and k? Explain. 10. If a pendulum clock keeps perfect time at the base of a mountain, will it also keep perfect time when it is moved to the top of the mountain? Explain. 11. You are looking at a small, leafy tree. You do not notice any breeze, and most of the leaves on the tree are motionless. One leaf, however, is fluttering back and forth wildly. After a while, that leaf stops moving and you notice a different leaf moving much more than all the others. Explain what could cause the large motion of one particular leaf. 12. A pendulum bob is made from a sphere filled with water. What would happen to the frequency of vibration of this pendulum if there were a hole in the sphere that allowed the water to leak out slowly? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 10 13. Consider the simplified single-piston engine in Figure CQ15.13. Assuming the wheel rotates with constant angular speed, explain why the piston rod oscillates in simple harmonic motion. Problems The problems found in this chapter may be assigned online in Enhanced WebAssign. 1. denotes straightforward problem; 2. denotes intermediate problem; 3. denotes challenging problem 1. full solution available in the Student Solutions Manual/ Study Guide 1. denotes problems most often assigned in Enhanced WebAssign; these provide students with targeted feedback and either a Master It tutorial or a Watch It solution video. Q|C denotes asking for quantitative and conceptual reasoning denotes symbolic reasoning problem denotes Master It tutorial available in Enhanced WebAssign Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 11 denotes guided problem shaded denotes "paired problems" that develop reasoning with symbols and numeric values Note: Ignore the mass of every spring, except in Problems 64 and 75. Section 15.1 Motion of an Object Attached to a Spring Problems 16, 17, 18, 22, and 65 in Chapter 7 can also be assigned with this section. 1. A 0.60-kg block attached to a spring with force constant 130 N/m is free to move on a frictionless, horizontal surface as in Active Figure 15.1. The block is released from rest when the spring is stretched 0.13 m. At the instant the block is released, find (a) the force on the block and (b) its acceleration. 2. When a 4.25-kg object is placed on top of a vertical spring, the spring compresses a distance of 2.62 cm. What is the force constant of the spring? Section 15.2 Analysis Model: Particle in Simple Harmonic Motion 3. A vertical spring stretches 3.9 cm when a 10-g object is hung Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 12 from it. The object is replaced with a block of mass 25 g that oscillates up and down in simple harmonic motion. Calculate the period of motion. 4. In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x = 5.00 cos 2t 6 where x is in centimeters and t is in seconds. At t = 0, find (a) the position of the particle, (b) its velocity, and (c) its acceleration. Find (d) the period and (e) the amplitude of the motion. 5. The position of a particle is given by the expression x = 4.00 cos (3.00t + ), where x is in meters and t is in seconds. Determine (a) the frequency and (b) period of the motion, (c) the amplitude of the motion, (d) the phase constant, and (e) the position of the particle at t = 0.250 s. 6. A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3 600 rev/min. Taking the extremes of its position relative to its center point as ±5.00 cm, find the magnitudes of the (a) maximum velocity and (b) maximum acceleration of the piston. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 13 7. A 1.00-kg object is attached to a horizontal spring. The spring is initially stretched by 0.100 m, and the object is released from rest there. It proceeds to move without friction. The next time the speed of the object is zero is 0.500 s later. What is the maximum speed of the object? 8. A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find (a) the period of its motion, (b) the frequency in hertz, and (c) the angular frequency in radians per second. 9. A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 s. Find the force constant of the spring. 10. Q|C (a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a moment 84.4 s later? (b) Find the distance traveled by the vibrating object in part (a). (c) What If? Another hanging spring stretches by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as x = 0. This object is also pulled down an additional Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 14 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later. (d) Find the distance traveled by the object in part (c). (e) Why are the answers to parts (a) and (c) so different when the initial data in parts (a) and (c) are so similar and the answers to parts (b) and (d) are relatively close? Does this circumstance reveal a fundamental difficulty in calculating the future? 11. Review. A particle moves along the x axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration 0.320 m/s2. Suppose it moves as a particle under constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume it moves as a particle in simple harmonic motion for 4.50 s and x = 0 is its equilibrium position. Find (c) its position and (d) its velocity at the end of this time interval. 12. Q|C A ball dropped from a height of 4.00 m makes an elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance, (a) show that the ensuing motion is periodic and (b) determine the period of the motion. (c) Is the motion simple harmonic? Explain. 13. A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 15 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Find an expression for the position of the particle as a function of time. Determine (b) the maximum speed of the particle and (c) the earliest time (t > 0) at which the particle has this speed. Find (d) the maximum positive acceleration of the particle and (e) the earliest time (t > 0) at which the particle has this acceleration. (f) Find the total distance traveled by the particle between t = 0 and t = 1.00 s. 14. A 1.00-kg glider attached to a spring with a force constant of 25.0 N/m oscillates on a frictionless, horizontal air track. At t = 0, the glider is released from rest at x = –3.00 cm (that is, the spring is compressed by 3.00 cm). Find (a) the period of the glider’s motion, (b) the maximum values of its speed and acceleration, and (c) the position, velocity, and acceleration as functions of time. 15. A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 10.0 cm. Calculate the maximum value of its (a) speed and (b) acceleration, (c) the speed and (d) the acceleration when the object is 6.00 cm from the equilibrium position, and (e) the time interval required for the object to move from x = 0 to x = 8.00 cm. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 16 16. Q|C You attach an object to the bottom end of a hanging vertical spring. It hangs at rest after extending the spring 18.3 cm. You then set the object vibrating. (a) Do you have enough information to find its period? (b) Explain your answer and state whatever you can about its period. Section 15.3 Energy of the Simple Harmonic Oscillator 17. To test the resiliency of its bumper during low-speed collisions, a 1 000-kg automobile is driven into a brick wall. The car’s bumper behaves like a spring with a force constant 5.00 106 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy is transformed or transferred away during impact with the wall? 18. A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. The total energy of the system is 2.00 J. Find (a) the force constant of the spring and (b) the amplitude of the motion. 19. A 50.0-g object connected to a spring with a force constant of 35.0 N/m oscillates with an amplitude of 4.00 cm on a frictionless, horizontal surface. Find (a) the total energy of the system and (b) the speed of the object when its position is 1.00 Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 17 cm. Find (c) the kinetic energy and (d) the potential energy when its position is 3.00 cm. 20. A 2.00-kg object is attached to a spring and placed on a frictionless, horizontal surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the object. (d) Where does this maximum speed occur? (e) Find the maximum acceleration of the object. (f) Where does the maximum acceleration occur? (g) Find the total energy of the oscillating system. Find (h) the speed and (i) the acceleration of the object when its position is equal to one-third the maximum value. 21. Q|C A simple harmonic oscillator of amplitude A has a total energy E. Determine (a) the kinetic energy and (b) the potential energy when the position is one-third the amplitude. (c) For what values of the position does the kinetic energy equal one-half the potential energy? (d) Are there any values of the position where the kinetic energy is greater than the maximum potential energy? Explain. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 18 22. Review. A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge. The unstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m free fall and a 25.0-m section of simple harmonic oscillation. (a) For the free- fall part, what is the appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or non- isolated? (d) From your response in part (c) find the spring constant of the bungee cord. (e) What is the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper? (f) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop? 23. Q|C Review. A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as in Figure P15.23. A horizontal force causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of . (b) What is the total Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 19 energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position? 24. A 326-g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 5.83 J, find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion. Section 15.4 Comparing Simple Harmonic Motion with Uniform Circular Motion 25. Q|C While driving behind a car traveling at 3.00 m/s, you Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 20 notice that one of the car’s tires has a small hemispherical bump on its rim as shown in Figure P15.25. (a) Explain why the bump, from your viewpoint behind the car, executes simple harmonic motion. (b) If the radii of the car’s tires are 0.300 m, what is the bump’s period of oscillation? Section 15.5 The Pendulum Problem 62 in Chapter 1 can also be assigned with this section. 26. A ―seconds pendulum‖ is one that moves through its equilibrium position once each second. (The period of the pendulum is precisely 2 s.) The length of a seconds pendulum is 0.992 7 m at Tokyo, Japan, and 0.994 2 m at Cambridge, England. What is the ratio of the free-fall accelerations at these two locations? 27. A simple pendulum makes 120 complete oscillations in 3.00 min at a location where g = 9.80 m/s2. Find (a) the period of the pendulum and (b) its length. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 21 28. A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R. That is, g / R . 29. A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency of 0.450 Hz. The pendulum has a mass of 2.20 kg, and the pivot is located 0.350 m from the center of mass. Determine the moment of inertia of the pendulum about the pivot point. 30. A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency f. The pendulum has a mass m, and the pivot is located a distance d from the center of mass. Determine the moment of inertia of the pendulum about the pivot point. 31. Q|C A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.0 and then released. Using the analysis model of a particle in simple harmonic motion, what are (a) the maximum speed of the bob, (b) its maximum angular acceleration, and (c) the maximum restoring force on the bob? (d) What If? Solve parts (a) through Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 22 (c) again by using analysis models introduced in earlier chapters. (e) Compare the answers. 32. Consider the physical pendulum of Figure 15.17. (a) Represent its moment of inertia about an axis passing through its center of mass and parallel to the axis passing through its pivot point as ICM. Show that its period is I CM md 2 T 2 mgd where d is the distance between the pivot point and the center of mass. (b) Show that the period has a minimum value when d satisfies md2 = ICM. 33. Review. A simple pendulum is 5.00 m long. What is the period of small oscillations for this pendulum if it is located in an elevator (a) accelerating upward at 5.00 m/s2? (b) Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 23 Accelerating downward at 5.00 m/s2? (c) What is the period of this pendulum if it is placed in a truck that is accelerating horizontally at 5.00 m/s2? 34. A very light rigid rod of length 0.500 m extends straight out from one end of a meterstick. The combination is suspended from a pivot at the upper end of the rod as shown in Figure P15.34. The combination is then pulled out by a small angle and released. (a) Determine the period of oscillation of the system. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long? 35. A watch balance wheel (Fig. P15.35) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around a rim of radius 0.500 cm. What are (a) the wheel’s moment of inertia and (b) the torsion constant of the attached spring? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 24 36. A small object is attached to the end of a string to form a simple pendulum. The period of its harmonic motion is measured for small angular displacements and three lengths. For lengths of 1.000 m, 0.750 m, and 0.500 m, total time intervals for 50 oscillations of 99.8 s, 86.6 s, and 71.1 s are measured with a stopwatch. (a) Determine the period of motion for each length. (b) Determine the mean value of g obtained from these three independent measurements and compare it with the accepted value. (c) Plot T2 versus L and obtain a value for g from the slope of your best-fit straight-line graph. (d) Compare the value found in part (c) with that obtained in part (b). Section 15.6 Damped Oscillations 37. A pendulum with a length of 1.00 m is released from an initial angle of 15.0. After 1 000 s, its amplitude has been Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 25 reduced by friction to 5.50. What is the value of b/2m? 38. Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = –bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E 1 mv2 1 kx2 , and 2 2 use Equation 15.31. 39. A 10.6-kg object oscillates at the end of a vertical spring that has a spring constant of 2.05 104 N/m. The effect of air resistance is represented by the damping coefficient b = 3.00 N · s/m. (a) Calculate the frequency of the damped oscillation. (b) By what percentage does the amplitude of the oscillation decrease in each cycle? (c) Find the time interval that elapses while the energy of the system drops to 5.00% of its initial value. 40. Show that Equation 15.32 is a solution of Equation 15.31 provided that b2 < 4mk. Section 15.7 Forced Oscillations 41. As you enter a fine restaurant, you realize that you have accidentally brought a small electronic timer from home instead of your cell phone. In frustration, you drop the timer Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 26 into a side pocket of your suit coat, not realizing that the timer is operating. The arm of your chair presses the light cloth of your coat against your body at one spot. Fabric with a length L hangs freely below that spot, with the timer at the bottom. At one point during your dinner, the timer goes off and a buzzer and a vibrator turn on and off with a frequency of 1.50 Hz. It makes the hanging part of your coat swing back and forth with remarkably large amplitude, drawing everyone’s attention. Find the value of L. 42. A baby bounces up and down in her crib. Her mass is 12.5 kg, and the crib mattress can be modeled as a light spring with force constant 700 N/m. (a) The baby soon learns to bounce with maximum amplitude and minimum effort by bending her knees at what frequency? (b) If she were to use the mattress as a trampoline—losing contact with it for part of each cycle— what minimum amplitude of oscillation does she require? 43. A 2.00-kg object attached to a spring moves without friction (b = 0) and is driven by an external force given by the expression F = 3.00 sin (2t), where F is in newtons and t is in seconds. The force constant of the spring is 20.0 N/m. Find (a) the resonance angular frequency of the system, (b) the angular frequency of the driven system, and (c) the amplitude of the motion. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 27 44. A block weighing 40.0 N is suspended from a spring that has a force constant of 200 N/m. The system is undamped (b = 0) and is subjected to a harmonic driving force of frequency 10.0 Hz, resulting in a forced-motion amplitude of 2.00 cm. Determine the maximum value of the driving force. 45. Damping is negligible for a 0.150-kg object hanging from a light, 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.440 m? 46. Considering an undamped, forced oscillator (b = 0), show that Equation 15.35 is a solution of Equation 15.34, with an amplitude given by Equation 15.36. Additional Problems 47. The mass of the deuterium molecule (D2) is twice that of the hydrogen molecule (H2). If the vibrational frequency of H2 is 1.30 1014 Hz, what is the vibrational frequency of D2? Assume the ―spring constant‖ of attracting forces is the same for the two molecules. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 28 48. Q|C Review. This problem extends the reasoning of Problem 59 in Chapter 9. Two gliders are set in motion on an air track. Glider 1 has mass m1 = 0.240 kg and moves to the right with speed 0.740 m/s. It will have a rear-end collision with glider 2, of mass m2 = 0.360 kg, which initially moves to the right with speed 0.120 m/s. A light spring of force constant 45.0 N/m is attached to the back end of glider 2 as shown in Figure P9.59. When glider 1 touches the spring, superglue instantly and permanently makes it stick to its end of the spring. (a) Find the common speed the two gliders have when the spring is at maximum compression. (b) Find the maximum spring compression distance. The motion after the gliders become attached consists of a combination of (1) the constant- velocity motion of the center of mass of the two-glider system found in part (a) and (2) simple harmonic motion of the gliders relative to the center of mass. (c) Find the energy of the center- of-mass motion. (d) Find the energy of the oscillation. 49. Q|C An object of mass m moves in simple harmonic motion with amplitude 12.0 cm on a light spring. Its maximum Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 29 acceleration is 108 cm/s2. Regard m as a variable. (a) Find the period T of the object. (b) Find its frequency f. (c) Find the maximum speed vmax of the object. (d) Find the total energy E of the object–spring system. (e) Find the force constant k of the spring. (f) Describe the pattern of dependence of each of the quantities T, f, vmax, E, and k on m. 50. Q|C Review. A rock rests on a concrete sidewalk. An earthquake strikes, making the ground move vertically in simple harmonic motion with a constant frequency of 2.40 Hz and with gradually increasing amplitude. (a) With what amplitude does the ground vibrate when the rock begins to lose contact with the sidewalk? Another rock is sitting on the concrete bottom of a swimming pool full of water. The earthquake produces only vertical motion, so the water does not slosh from side to side. (b) Present a convincing argument that when the ground vibrates with the amplitude found in part (a), the submerged rock also barely loses contact with the floor of the swimming pool. 51. A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P15.51). Determine the tensions in the rod (a) at the pivot and (b) at the point P when the system is stationary. (c) Calculate the period of oscillation for small displacements from Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 30 equilibrium and (d) determine this period for L = 2.00 m. 52. An object attached to a spring vibrates with simple harmonic motion as described by Figure P15.52. For this motion, find (a) the amplitude, (b) the period, (c) the angular frequency, (d) the maximum speed, (e) the maximum acceleration, and (f) an equation for its position x as a function of time. 53. Review. A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.50 Hz. Block B rests on it as shown in Figure P15.53, and the coefficient of static Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 31 friction between the two is s = 0.600. What maximum amplitude of oscillation can the system have if block B is not to slip? 54. Review. A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f. Block B rests on it as shown in Figure P15.53, and the coefficient of static friction between the two is s. What maximum amplitude of oscillation can the system have if block B is not to slip? 55. A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig. P15.55). Find the frequency of vibration of the system for small values of the amplitude (small ). Assume the vertical suspension rod of length L is rigid, but ignore its mass. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 32 56. A particle with a mass of 0.500 kg is attached to a horizontal spring with a force constant of 50.0 N/m. At the moment t = 0, the particle has its maximum speed of 20.0 m/s and is moving to the left. (a) Determine the particle’s equation of motion, specifying its position as a function of time. (b) Where in the motion is the potential energy three times the kinetic energy? (c) Find the minimum time interval required for the particle to move from x = 0 to x = 1.00 m. (d) Find the length of a simple pendulum with the same period. 57. A horizontal plank of mass 5.00 kg and length 2.00 m is pivoted at one end. The plank’s other end is supported by a spring of force constant 100 N/m (Fig P15.57). The plank is displaced by a small angle from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 33 58. A horizontal plank of mass m and length L is pivoted at one end. The plank’s other end is supported by a spring of force constant k (Fig P15.57). The plank is displaced by a small angle from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion. 59. Q|C Review. A particle of mass 4.00 kg is attached to a spring with a force constant of 100 N/m. It is oscillating on a frictionless, horizontal surface with an amplitude of 2.00 m. A 6.00-kg object is dropped vertically on top of the 4.00-kg object as it passes through its equilibrium point. The two objects stick together. (a) What is the new amplitude of the vibrating system after the collision? (b) By what factor has the period of the system changed? (c) By how much does the energy of the system change as a result of the collision? (d) Account for the change in energy. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 34 60. A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine (a) its period, (b) its total energy, and (c) its maximum angular displacement. 61. Review. One end of a light spring with force constant k = 100 N/m is attached to a vertical wall. A light string is tied to the other end of the horizontal spring. As shown in Figure P15.61, the string changes from horizontal to vertical as it passes over a pulley of mass M in the shape of a solid disk of radius R = 2.00 cm. The pulley is free to turn on a fixed, smooth axle. The vertical section of the string supports an object of mass m = 200 g. The string does not slip at its contact with the pulley. The object is pulled downward a small distance and released. (a) What is the angular frequency of oscillation of the object in terms of the mass M? (b) What is the highest possible value of the angular frequency of oscillation of the object? (c) What is the highest possible value of the angular frequency of oscillation of the object if the pulley radius is doubled to R = 4.00 cm? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 35 62. Q|C People who ride motorcycles and bicycles learn to look out for bumps in the road and especially for washboarding, a condition in which many equally spaced ridges are worn into the road. What is so bad about washboarding? A motorcycle has several springs and shock absorbers in its suspension, but you can model it as a single spring supporting a block. You can estimate the force constant by thinking about how far the spring compresses when a heavy rider sits on the seat. A motorcyclist traveling at highway speed must be particularly careful of washboard bumps that are a certain distance apart. What is the order of magnitude of their separation distance? 63. A ball of mass m is connected to two rubber bands of length L, each under tension T as shown in Figure P15.63. The ball is displaced by a small distance y perpendicular to the length of the rubber bands. Assuming the tension does not change, show that (a) the restoring force is – (2T/L) y and (b) the system exhibits simple harmonic motion with an angular frequency 2T / mL . Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 36 64. When a block of mass M, connected to the end of a spring of mass ms = 7.40 g and force constant k, is set into simple harmonic motion, the period of its motion is M ms / 3 T 2 k A two-part experiment is conducted with the use of blocks of various masses suspended vertically from the spring as shown in Figure P15.64. (a) Static extensions of 17.0, 29.3, 35.3, 41.3, 47.1, and 49.3 cm are measured for M values of 20.0, 40.0, 50.0, 60.0, 70.0, and 80.0 g, respectively. Construct a graph of Mg versus x and perform a linear least-squares fit to the data. (b) From the slope of your graph, determine a value for k for this spring. (c) The system is now set into simple harmonic motion, and periods are measured with a stopwatch. With M = 80.0 g, the total time interval required for ten oscillations is measured to be 13.41 s. The experiment is repeated with M values of 70.0, 60.0, 50.0, 40.0, and 20.0 g, with corresponding time intervals for ten oscillations of 12.52, 11.67, 10.67, 9.62, and 7.03 s. (d) Compute the experimental value for T from each of these Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 37 measurements. (e) Plot a graph of T2 versus M and (f) determine a value for k from the slope of the linear least- squares fit through the data points. (g) Compare this value of k with that obtained in part (b). (h) Obtain a value for ms from your graph and compare it with the given value of 7.40 g. 65. Review. A light balloon filled with helium of density 0.179 kg/m3 is tied to a light string of length L = 3.00 m. The string is tied to the ground forming an ―inverted‖ simple pendulum (Fig. 15.65a). If the balloon is displaced slightly from equilibrium as in Figure P15.65b and released, (a) show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.20 kg/m3. Hint: Use an analogy with the simple pendulum and see Chapter 14. Assume the air applies a buoyant force on the balloon but does not otherwise affect its motion. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 38 66. Consider the damped oscillator illustrated in Figure 15.20. The mass of the object is 375 g, the spring constant is 100 N/m, and b = 0.100 N · s/m. (a) Over what time interval does the amplitude drop to half its initial value? (b) What If? Over what time interval does the mechanical energy drop to half its initial value? (c) Show that, in general, the fractional rate at which the amplitude decreases in a damped harmonic oscillator is one- half the fractional rate at which the mechanical energy decreases. 67. A block of mass m is connected to two springs of force constants k1 and k2 in two ways as shown in Figure P15.67. In both cases, the block moves on a frictionless table after it is displaced from equilibrium and released. Show that in the two cases the block exhibits simple harmonic motion with periods m k1 k2 m (a) T 2 and (b) T 2 k1k2 k1 k2 Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 39 68. Your thumb squeaks on a plate you have just washed. Your sneakers squeak on the gym floor. Car tires squeal when you start or stop abruptly. You can make a goblet sing by wiping your moistened finger around its rim. When chalk squeaks on a blackboard, you can see that it makes a row of regularly spaced dashes. As these examples suggest, vibration commonly results when friction acts on a moving elastic object. The oscillation is not simple harmonic motion, but is called stick-and-slip. This problem models stick-and-slip motion. A block of mass m is attached to a fixed support by a horizontal spring with force constant k and negligible mass (Fig. P15.68). Hooke’s law describes the spring both in extension and in compression. The block sits on a long horizontal board, with which it has coefficient of static friction s and a smaller coefficient of kinetic friction k. The board moves to the right at constant speed v. Assume the block spends most of its time sticking to the board and moving to the Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 40 right with it, so the speed v is small in comparison to the average speed the block has as it slips back toward the left. (a) Show that the maximum extension of the spring from its unstressed position is very nearly given by smg/k. (b) Show that the block oscillates around an equilibrium position at which the spring is stretched by kmg/k. (c) Graph the block’s position versus time. (d) Show that the amplitude of the block’s motion is A s k mg k (e) Show that the period of the block’s motion is 2 s k mg m T vk k It is the excess of static over kinetic friction that is important for the vibration. ―The squeaky wheel gets the grease‖ because even a viscous fluid cannot exert a force of static friction. 69. Review. A lobsterman’s buoy is a solid wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm seawater, having density . A passing shark tugs on the slack rope mooring the buoy to a Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 41 lobster trap, pulling the buoy down a distance x from its equilibrium position and releasing it. (a) Show that the buoy will execute simple harmonic motion if the resistive effects of the water are ignored. (b) Determine the period of the oscillations. 70. Why is the following situation impossible? Your job involves building very small damped oscillators. One of your designs involves a spring–object oscillator with a spring of force constant k = 10.0 N/m and an object of mass m = 1.00 g. Your design objective is that the oscillator undergo many oscillations as its amplitude falls to 25.0% of its initial value in a certain time interval. Measurements on your latest design show that the amplitude falls to the 25.0% value in 23.1 ms. This time interval is too long for what is needed in your project. To shorten the time interval, you double the damping constant b for the oscillator. This doubling allows you to reach your design objective. 71. Two identical steel balls, each of mass 67.4 g, are moving in opposite directions at 5.00 m/s. They collide head-on and bounce apart elastically. By squeezing one of the balls in a vise while precise measurements are made of the resulting amount of compression, you find that Hooke’s law is a good model of the ball’s elastic behavior. A force of 16.0 kN exerted by each Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 42 jaw of the vise reduces the diameter by 0.200 mm. Model the motion of each ball, while the balls are in contact, as one-half of a cycle of simple harmonic motion. Compute the time interval for which the balls are in contact. (If you solved Problem 57 in Chapter 7, compare your results from this problem with your results from that one.) Challenge Problems 72. A smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P15.72. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is 1/2 Rg 1 cos v 2 M / m r / R 2 2 (b) Show that the period of the motion is M 2m R 2 mr 2 1/2 T 2 2mgR Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 43 73. An object of mass m1 = 9.00 kg is in equilibrium when connected to a light spring of constant k = 100 N/m that is fastened to a wall as shown in Figure P15.73a. A second object, m2 = 7.00 kg, is slowly pushed up against m1, compressing the spring by the amount A = 0.200 m (see Fig. P15.73b). The system is then released, and both objects start moving to the right on the frictionless surface. (a) When m1 reaches the equilibrium point, m2 loses contact with m1 (see Fig. P15.73c) and moves to the right with speed v. Determine the value of v. (b) How far apart are the objects when the spring is fully stretched for the first time (the distance D in Fig. P15.48d)? Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 44 74. Review. Why is the following situation impossible? You are in the high-speed package delivery business. Your competitor in the next building gains the right-of-way to build an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth. By simply dropping packages into this tunnel, they fall downward and arrive at the other end of your tunnel, which is Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 45 in a building right next to the other end of your competitor’s tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.74). Assume the Earth has uniform density. 75. A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a frictionless, horizontal track (Fig. P15.75). The force constant of the spring is k, and the equilibrium length is . Assume all portions of the spring oscillate in phase and the velocity of a segment of the spring of length dx is proportional to the distance x from the fixed end; that is, vx = (x/)v. Also, notice that the mass of a segment of the spring is dm = (m/) dx. Find (a) the kinetic energy of the system when the block has a speed v and (b) the period of oscillation. Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 46 76. Review. A system consists of a vertical spring with force constant k = 1 250 N/m, length L = 1.50 m, and an object of mass m = 5.00 kg attached to the end (Fig. P15.76). The object is placed at the level of the point of attachment with the spring unstretched, at position yi = L, and then it is released so that it swings like a pendulum. (a) Find the y position of the object at the lowest point. (b) Will the pendulum’s period be greater or less than the period of a simple pendulum with the same mass m and length L? Explain. 77. A light, cubical container of volume a3 is initially filled with a liquid of mass density as shown in Figure P15.77a. The cube is initially supported by a light string to form a simple pendulum of length Li, measured from the center of mass of the Serway/Jewett: PSE 8e Problems Set – Ch. 15 - 47 filled container, where Li >> a. The liquid is allowed to flow from the bottom of the container at a constant rate (dM/dt). At any time t, the level of the liquid in the container is h and the length of the pendulum is L (measured relative to the instantaneous center of mass) as shown in Figure P15.77b. (a) Find the period of the pendulum as a function of time. (b) What is the period of the pendulum after the liquid completely runs out of the container?
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